HAMILTONIAN-PRESERVING DISCONTINUOUS GALERKIN METHODS FOR THE LIOUVILLE EQUATION WITH DISCONTINUOUS POTENTIAL.

Numerically solving the Liouville equation in classical mechanics with a discontinuous potential often leads to the challenges of how to preserve the Hamiltonian across the potential barrier and a severe time step constraint according to the Courant--Friedrichs--Lewy condition. Motivated by the Hamiltonian-preserving finite volume schemes by Jin and Wen [21], we introduce a Hamiltonian-preserving discontinuous Galerkin (DG) scheme for the Liouville equation with discontinuous potential in this paper. The DG method can be designed with arbitrary order of accuracy and offers many advantages including easy adaptivity, compact stencils, and the ability of handling complicated boundary conditions and interfaces. We propose to carefully design the numerical fluxes of the DG methods to build the behavior of a classical particle at the potential barrier into the numerical scheme, which ensures the continuity of the Hamiltonian across the potential barrier and the correct transmission and reflection condition. Our scheme is proved to be positive and stable in L¹ norm if the positivity-preserving limiter is applied. Numerical examples are provided to illustrate the accuracy and effectiveness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]